Задание 36. Сравните значения выражений:

Решение:

1. \( 2\sqrt{7} \) и \( \sqrt{7} \cdot 2\sqrt{8} \)

\( 2\sqrt{7} = \sqrt{4}\sqrt{7} = \sqrt{28}\)

\( \sqrt{7} \cdot 2\sqrt{8} = \sqrt{7} \cdot \sqrt{4}\sqrt{8} = \sqrt{7} \cdot \sqrt{32} = \sqrt{224}\)

\( \sqrt{28} < \sqrt{224} \), значит, \( 2\sqrt{7} < \sqrt{7} \cdot 2\sqrt{8} \).

2. \( 3\sqrt{6} \) и \( \sqrt{60} \)

\( 3\sqrt{6} = \sqrt{9}\sqrt{6} = \sqrt{54}\)

\( \sqrt{54} < \sqrt{60} \), значит, \( 3\sqrt{6} < \sqrt{60} \).

3. \( 2\sqrt{6} \) и \( \sqrt{21} \)

\( 2\sqrt{6} = \sqrt{4}\sqrt{6} = \sqrt{24}\)

\( \sqrt{24} > \sqrt{21} \), значит, \( 2\sqrt{6} > \sqrt{21} \).

4. \( 3\sqrt{4} \) и \( 6 \)

\( 3\sqrt{4} = 3 \cdot 2 = 6\)

\( 6 = 6 \), значит, \( 3\sqrt{4} = 6 \).

5. \( \sqrt{40} \) и \( 2\sqrt{11} \)

\( 2\sqrt{11} = \sqrt{4}\sqrt{11} = \sqrt{44}\)

\( \sqrt{40} < \sqrt{44} \), значит, \( \sqrt{40} < 2\sqrt{11} \).

6. \( 4\sqrt{2} \) и \( \sqrt{30} \)

\( 4\sqrt{2} = \sqrt{16}\sqrt{2} = \sqrt{32}\)

\( \sqrt{32} > \sqrt{30} \), значит, \( 4\sqrt{2} > \sqrt{30} \).

7. \( 4\sqrt{5} \) и \( \sqrt{80} \)

\( 4\sqrt{5} = \sqrt{16}\sqrt{5} = \sqrt{80}\)

\( \sqrt{80} = \sqrt{80} \), значит, \( 4\sqrt{5} = \sqrt{80} \).

8. \( 2\sqrt{7} \) и \( 5 \)

\( 2\sqrt{7} = \sqrt{4}\sqrt{7} = \sqrt{28}\)

\( 5 = \sqrt{25}\)

\( \sqrt{28} > \sqrt{25} \), значит, \( 2\sqrt{7} > 5 \).

9. \( 6\sqrt{3} \) и \( 10 \)

\( 6\sqrt{3} = \sqrt{36}\sqrt{3} = \sqrt{108}\)

\( 10 = \sqrt{100}\)

\( \sqrt{108} > \sqrt{100} \), значит, \( 6\sqrt{3} > 10 \).

10. \( 3\sqrt{3} \) и \( \sqrt{10} \)

\( 3\sqrt{3} = \sqrt{9}\sqrt{3} = \sqrt{27}\)

\( \sqrt{27} > \sqrt{10} \), значит, \( 3\sqrt{3} > \sqrt{10} \).

11. \( 2\sqrt{3} \) и \( 3\sqrt{2} \)

\( 2\sqrt{3} = \sqrt{4}\sqrt{3} = \sqrt{12}\)

\( 3\sqrt{2} = \sqrt{9}\sqrt{2} = \sqrt{18}\)

\( \sqrt{12} < \sqrt{18} \), значит, \( 2\sqrt{3} < 3\sqrt{2} \).

12. \( 8\sqrt{2} \) и \( 8\sqrt{6} \)

\( 8\sqrt{2} = \sqrt{64}\sqrt{2} = \sqrt{128}\)

\( 8\sqrt{6} = \sqrt{64}\sqrt{6} = \sqrt{384}\)

\( \sqrt{128} < \sqrt{384} \), значит, \( 8\sqrt{2} < 8\sqrt{6} \).

13. \( -2\sqrt{6} \) и \( -8\sqrt{2} \)

\( -2\sqrt{6} = -\sqrt{4}\sqrt{6} = -\sqrt{24}\)

\( -8\sqrt{2} = -\sqrt{64}\sqrt{2} = -\sqrt{128}\)

\( -\sqrt{24} > -\sqrt{128} \), значит, \( -2\sqrt{6} > -8\sqrt{2} \).

14. \( 5\sqrt{3} \) и \( 6\sqrt{2} \)

\( 5\sqrt{3} = \sqrt{25}\sqrt{3} = \sqrt{75}\)

\( 6\sqrt{2} = \sqrt{36}\sqrt{2} = \sqrt{72}\)

\( \sqrt{75} > \sqrt{72} \), значит, \( 5\sqrt{3} > 6\sqrt{2} \).

15. \( 3\sqrt{10} \) и \( 7\sqrt{2} \)

\( 3\sqrt{10} = \sqrt{9}\sqrt{10} = \sqrt{90}\)

\( 7\sqrt{2} = \sqrt{49}\sqrt{2} = \sqrt{98}\)

\( \sqrt{90} < \sqrt{98} \), значит, \( 3\sqrt{10} < 7\sqrt{2} \).

16. \( -10\sqrt{6} \) и \( -5\sqrt{10} \)

\( -10\sqrt{6} = -\sqrt{100}\sqrt{6} = -\sqrt{600}\)

\( -5\sqrt{10} = -\sqrt{25}\sqrt{10} = -\sqrt{250}\)

\( -\sqrt{600} < -\sqrt{250} \), значит, \( -10\sqrt{6} < -5\sqrt{10} \).

17. \( -2\sqrt{8} \) и \( -4\sqrt{2} \)

\( -2\sqrt{8} = -\sqrt{4}\sqrt{8} = -\sqrt{32}\)

\( -4\sqrt{2} = -\sqrt{16}\sqrt{2} = -\sqrt{32}\)

\( -\sqrt{32} = -\sqrt{32} \), значит, \( -2\sqrt{8} = -4\sqrt{2} \).

18. \( -2\sqrt{1,1} \) и \( -2\sqrt{1,5} \)

\( -2\sqrt{1,1} = -\sqrt{4}\sqrt{1,1} = -\sqrt{4,4}\)

\( -2\sqrt{1,5} = -\sqrt{4}\sqrt{1,5} = -\sqrt{6}\)

\( -\sqrt{4,4} > -\sqrt{6} \), значит, \( -2\sqrt{1,1} > -2\sqrt{1,5} \).

19. \( 10\sqrt{20} \) и \( 20\sqrt{10} \)

\( 10\sqrt{20} = \sqrt{100}\sqrt{20} = \sqrt{2000}\)

\( 20\sqrt{10} = \sqrt{400}\sqrt{10} = \sqrt{4000}\)

\( \sqrt{2000} < \sqrt{4000} \), значит, \( 10\sqrt{20} < 20\sqrt{10} \).

20. \( -0,2\sqrt{5} \) и \( -0,5\sqrt{2} \)

\( -0,2\sqrt{5} = -\sqrt{0,04}\sqrt{5} = -\sqrt{0,2}\)

\( -0,5\sqrt{2} = -\sqrt{0,25}\sqrt{2} = -\sqrt{0,5}\)

\( -\sqrt{0,2} > -\sqrt{0,5} \), значит, \( -0,2\sqrt{5} > -0,5\sqrt{2} \).